On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups

TL;DR

This paper completes the study of three-dimensional Dirichlet stereohedra, establishing an upper bound of 92 facets for all crystallographic groups, using classification and computational methods.

Contribution

It extends previous bounds to the remaining quarter cubic groups, providing a comprehensive upper limit for Dirichlet stereohedra facets across all 3D crystallographic groups.

Findings

Dirichlet stereohedra for quarter cubic groups have at most 92 facets.

The study confirms bounds for all three-dimensional crystallographic groups.

Computer-assisted methods were crucial for the analysis.

Abstract

In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by the second author and D. Bochis, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 "full" cubic groups cannot have more than 25 facets. Here we study the remaining "quarter" cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets.